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ZZ Instantons and the Non-Perturbative Dual
of c = 1 String Theory
Bruno Balthazar, Victor A. Rodriguez, Xi Yin
Jefferson Physical Laboratory, Harvard University,
Cambridge, MA 02138 USA
[email protected], [email protected],
Abstract
We study the effect of ZZ instantons in c = 1 string theory, and demonstrate that
they give rise to non-perturbative corrections to scattering amplitudes that do not
saturate unitarity within the closed string sector. Beyond the leading non-perturbative
order, logarithmic divergences are canceled between worldsheet diagrams of different
topologies, due to the Fischler-Susskind-Polchinski mechanism. We propose that the
closed string vacuum in c = 1 string theory is non-perturbatively dual to a state of
the matrix quantum mechanics in which all scattering states up to a given energy with
no incoming flux from the “other side” of the potential are occupied by free fermions.
Under such a proposal, we find detailed agreement of non-perturbative corrections to
closed string amplitudes in the worldsheet description and in the dual matrix model.
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Contents
1 Introduction 1
2 Closed string amplitudes mediated by the ZZ instanton 4
2.1 The structure of D-instanton expansion and the Fischler-Susskind-Polchinki
mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Leading correction to the 1→ n closed string amplitude . . . . . . . . . . . 7
2.3 Next-to-leading order correction to the 1→ 1 amplitude . . . . . . . . . . . 8
2.3.1 Disc 2-point diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Annulus 1-point diagram . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.3 Integration over collective coordinate and cancelation of divergences . 14
3 Comparison with the matrix model dual 15
4 Discussion 19
A The t→ 0 limit of the annulus 1-point diagram 21
B Some numerical details 22
B.1 Disc 2-point diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
B.2 Annulus 1-point diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1 Introduction
The c = 1 string theory [1–5] describes bosonic strings propagating in two spacetime dimen-
sions, whose worldsheet theory consists of a free time-like boson X0 and a c = 25 Liouville
theory, together with the bc ghost system. Much of the interest in the subject has to do
with the duality between c = 1 string theory and a large N gauged matrix quantum me-
chanics, which may be reformulated as the theory of a large number of free non-relativistic
fermions subject to the Hamiltonian H = 12p2 − 1
2x2, near a state in which the fermions fill
the region H < −µ, x > 0 in the phase space. The closed strings are dual to collective exci-
tations of the fermi surface. The agreement of perturbative scattering amplitudes computed
in the worldsheet theory and in the matrix model has been explicitly demonstrated at tree
1
level [6–9], and at 1-loop order [9]. This duality has been extended to the gauge non-singlet
sector of the matrix model [10, 11], whose states are mapped to “long strings”, defined as
high energy limit of open strings attached to FZZT branes [12, 13] that reside in the weak
coupling region.
Conventionally, the above mentioned duality is thought to hold only at the level of
perturbation theory, as the fermi sea is subject to the non-perturbative instability due to
the fermion tunneling to the “other side” (x < 0) of the potential. It was proposed in [14]
that a fermion near the top of the potential is dual to the (unstable) ZZ-brane [15]. The open
string tachyon excitation on the ZZ-brane allows it to access either side of the potential. This
hints that string perturbation theory augmented by D-branes may give a non-perturbative
formulation of the c = 1 string.
In this paper, we find evidence that the worldsheet formulation of c = 1 string theory
can be completed non-perturbatively by taking into account D-instanton effects. Specifically,
we consider ZZ-instantons, defined as a boundary condition on the worldsheet of Dirichlet
type in X0 direction and of ZZ type in the Liouville CFT, which are expected to be dual
to the “bounce” solution [16] describing the tunneling of a single fermion through the po-
tential barrier. We will study the non-perturbative correction to the closed string 1 → n
reflection amplitude mediated by a single ZZ-instanton, which renders the amplitude non-
unitary. The leading contribution, of order e−SZZ , comes from n+ 1 disconnected discs, each
emitting/absorbing a closed string, all of which have their boundaries ending on the same
ZZ-instanton. In the n = 1 case, we further analyze the next-to-leading order contribution,
of order e−SZZgs, coming from (i) a single disc with two closed string insertions, and (ii) a
disconnected diagram consisting of a disc and an annulus, each emitting/absorbing a closed
string, such that divergences in the open string channel cancel between (i) and (ii) due to
the Fischler-Susskind-Polchinski mechanism [17–20].
V+ω
V−ω1 . . .V−ωn
Figure 1: Worldsheet diagram that computes the leading non-perturbative correction to the
closed string 1→ n amplitude, of order e−SZZ , mediated by a single ZZ-instanton.
Our results for the leading one-instanton 1→ n amplitude and the next-to-leading order
one-instanton 1 → 1 amplitude agree, in a striking manner, with the non-perturbative
correction to the reflection amplitude of a collective field in the dual matrix model, assuming
that the fermi sea consists of only fermions that occupy scattering states with no incoming
2
V+ω1
V−ω2+
V+ω1
V−ω2+
V+ω1
V−ω2
Figure 2: Worldsheet diagrams that compute the next-to-leading order non-perturbative
correction to the closed string 1 → 1 amplitude. The logarithmic divergences in the open
string channel cancel between the connected and disconnected diagrams, as an example of
the Fischler-Susskind-Polchinski mechanism.
flux from the other side of the potential. We conjecture that the the latter is the correct
dual of the closed string vacuum of the non-perturbative completion of c = 1 string theory.
Figure 3: The closed string vacuum is dual to the Fermi sea of scattering states with no
incoming flux from the left side of the potential.
The sense in which the “non-perturbative corrections” are well-defined requires some
explanation. While the perturbative closed string amplitudes of c = 1 string theory obey
perturbative unitarity, the ZZ-instanton effects generally renders the amplitude non-unitary
within the closed string sector. The transition probability of closed string into other types
of asymptotic states is unambiguously defined and is captured entirely by non-perturbative
corrections. This is not the end of the story, however. The perturbative expansion of the
closed c = 1 string amplitude, as seen from the matrix model result, is an asymptotic series in
gs that is Borel summable. According to our proposal, there are non-perturbative corrections
to the Borel-resummed perturbative amplitude, and such corrections are captured by ZZ-
instanton effects.
The unitary completion of the amplitude would require considering individual fermions
on the other side of the potential among the asymptotic states, which are dual to rolling open
3
string tachyons on the ZZ-brane [21,22]. A detailed treatment of such states in the scattering
problem is beyond the scope of this paper. Let us also emphasize that our proposal for the
non-perturbative completion of c = 1 string theory is not equivalent to the type 0B string
theory/matrix model [23, 24]. In the latter setting, the fermi sea fills in both sides of the
potential, and the worldsheet theory involves an N = 1 superconformal field theory coupled
to superconformal ghosts, subject to diagonal GSO projection, rather than the bosonic theory
as considered here.
This paper is organized as follows. In section 2 we outline the structure of ZZ-instanton
contributions to closed string scattering amplitudes, and present explicit one-instanton com-
putations at the leading and next-to-leading order in gs. The results are then compared to
the proposed matrix model dual in section 3. After fixing certain finite counter terms (in
the sense of open+closed string effective field theory on the ZZ-instanton), we find highly
nontrivial agreement. Some future prospectives are discussed in section 4.
2 Closed string amplitudes mediated by the ZZ instan-
ton
The ZZ instanton is defined from the worldsheet perspective as a boundary condition that is
of Neumann type in bc ghosts, Dirichlet in the X0 CFT, and ZZ boundary condition in the
Liouville CFT. Its action SZZ is related to the mass MZZ = µ = 12πgs
of the ZZ-brane [14] (in
α′ = 1 units) by [25]
SZZ = 2πMZZ =1
gs. (2.1)
This is equal to the action of the bounce solution [16] that computes the semi-classical
tunneling probability e−Sbounce of a fermion at the fermi surface H = −µ through the potential
barrier,
Sbounce = 2
∫ √2µ
−√
2µ
dx√
2µ− x2 = 2πµ. (2.2)
We expect closed string amplitudes mediated by ZZ-instantons to be non-perturbative cor-
rections that render the closed string S-matrix Sc non-unitary due to tunneling effects. In
particular, 1−S†cSc computes the probability of closed string states turning into other kinds
of asymptotic states, such as ZZ-branes with rolling open string tachyon.
4
2.1 The structure of D-instanton expansion and the Fischler-Susskind-
Polchinki mechanism
The structure of the ZZ-instanton contribution to closed string amplitudes, as a special case
of D-instanton amplitudes, was outlined in [17]. The one-instanton contribution to a closed
string n-point connected amplitude, for instance, is computed by worldsheet diagrams with
boundary components along which the string ends on the same ZZ-instanton, integrated over
the collective coordinate of the ZZ-instanton. Note that, even though we are speaking of a
contribution to the connected closed string S-matrix element, the worldsheet diagram need
not be connected.
Let us first discuss a connected component of the worldsheet diagram with no closed
string insertion. The “empty” disc diagram with no closed string insertion represents −SZZ.
Formally, the summation over multiple disconnected discs, with symmetry factor taken into
account, exponentiates to e−SZZ . With this understanding, we will simply drop all discon-
nected empty discs and include an overall factor of e−SZZ in the one-instanton amplitude.
Connected components that are empty diagrams with non-negative Euler characteristic
amount to quantum corrections (renormalization) of the ZZ-instanton action SZZ, which
are suppressed by powers of gs. After taking this into account, the leading one-instanton
contribution to the closed string amplitude comes from n disconnected discs, each with a
closed string insertion, of the form1
e−SZZ
∫dx0
n∏i=1
〈Vi〉D2
ZZ,x0 , (2.3)
where 〈Vi〉D2
ZZ,x0 is the disc amplitude of a single closed string vertex operator Vi, with the
appropriate ghost insertions, whose boundary condition is that of the ZZ-instanton. In
particular, if Vi has energy ωi, e.g. Vi = gseiωiX
0VPi=
|ωi|2
(VP being a primary of the Liouville
CFT labeled by Liouville momentum P ), then 〈Vi〉D2
ZZ,x0 depends on the collective coordinate
x0 through the factor eiωix0. The integration over x0 then produces the delta function
δ(∑
i ωi) that imposes energy conservation. The resulting contribution to the n-point closed
string amplitude is of order e−1/gs , with no extra factors of gs, as each disc comes with a
factor of 1/gs that cancels against the gs from the vertex operator.
At the next order in gs, there will be contributions from the disc diagram with two
closed string insertions, as well as the annulus diagram with one closed string insertion. For
instance, there is a contribution to the 1→ 1 amplitude at order e−1gs gs of the form
e−SZZ
∫dx0
(〈V1V2〉D
2
ZZ,x0 + 〈V1〉D2
ZZ,x0 〈V2〉A2
ZZ,x0 + 〈V2〉D2
ZZ,x0 〈V1〉A2
ZZ,x0
), (2.4)
1This was already noted in the computation of D-instanton effects in type IIB superstring theory in [20].
5
where 〈Vi〉A2
ZZ,x0 is the annulus amplitude with a single closed string vertex operator Vi, whose
both boundaries end on the ZZ-instanton at time x0.
Each of the diagrams appearing in (2.4) suffers from divergence due to integration near
the boundary of the moduli space, where a long/thin strip develops (Figure 4). There is a
“tachyonic” open string mode on the ZZ-instanton, whose propagation through the strip gives
rise to a power divergence, which can be regularized by a simple subtraction as usual. On
the other hand, the “massless” open string mode on the ZZ-instanton, corresponding to the
collective coordinate, gives rise to a logarithmic divergence. Such logarithmic divergences,
as explained in [17,20], cancel between diagrams of different topologies in (2.4).
V+ω1
V−ω2
V+ω1
V−ω2V+ω1
V−ω2
Figure 4: Degeneration limit of the diagrams in Figure 2 that leads to logarithmic divergences
due to propagation of the open string collective mode on the ZZ-instanton. Such divergences
cancel in the sum of the three diagrams, as an example of the Fischler-Susskind-Polchinski
mechanism.
In carrying out this cancelation mechanism, one must first regularize the logarithmic
divergences by cutting off near the boundary of the moduli space, for each diagram in (2.4).
There is a finite ambiguity in choosing the cutoff between diagrams of different topologies,
proportional to
e−SZZ
∫dx0 ∂x0 〈V1〉D
2
ZZ,x0 ∂x0 〈V2〉D2
ZZ,x0 . (2.5)
From the point of view of closed+open string field theory, where the open string fields live
on the ZZ-instanton, this ambiguity is associated with a possible counter term in the open-
closed string effective vertex that is compatible with the Batalin-Vilkovisky master equation.
6
We do not know an a priori way to fix the constant coefficient of the ambiguity (2.5) in the
worldsheet computation of ZZ-instanton amplitudes. It will, however, be fixed through a
comparison with the dual matrix model.
2.2 Leading correction to the 1→ n closed string amplitude
The 1→ n S-matrix element of c = 1 string theory takes the form
S1→n(ω;ω1, · · · , ωn) = δ
(ω −
n∑i=1
ωi
)A1→n(ω1, · · · , ωn), (2.6)
where ω is the energy of the incoming closed string, and ω1, · · · , ωn are the energies of the
reflected/outgoing closed strings. The unitarity relation takes the form∫ωi≥0,
∑nj=1 ωj=ω
n∏i=1
dωi|A1→n(ω1, · · · , ωn)|2
ωω1 · · ·ωn= 1− P, (2.7)
where P is the probability of a closed string scattering into non-perturbative states.
The leading correction to A1→n that violates unitarity within the closed string sector,
thereby contributing to P , is due to the effect of a single ZZ-instanton, of order e−1/gs . The
relevant worldsheet diagram consists of n + 1 disconnected discs, each with a closed string
vertex operator insertion and with boundary ending on the same ZZ-instanton, as shown in
Figure 1.
The disc diagram with a single closed string vertex operator V±ω = gse±iωX0
Vω2
insertion,
corresponding to either an in-state or an out-state, is given by⟨V±ω (z, z)
⟩D2
ZZ,x0= gs
CD2
2πe±iωx
0
ΨZZ(ω
2
)= 2e±iωx
0
sinh(πω), (2.8)
where x0 is the location of the ZZ-instanton in time, and ΨZZ is the disc 1-point function of
the ZZ boundary state in Liouville CFT,
ΨZZ(P ) = 254√π sinh(2πP ). (2.9)
We have omitted writing the ghost insertions on the LHS. The factor of 12π
after the first
equality is due to division by the volume of the residual conformal Killing group. The
normalization constant CD2 associated with the disc topology was determined in [11], and
is given by
CD2 =2
34√π
gs. (2.10)
7
The leading one-ZZ-instanton contribution to S1→n(ω;ω1, · · · , ωn) is then evaluated as
Sinst,(0)1→n = N
∫ ∞−∞
dx0e−SZZ⟨V+ω (0)
⟩D2
ZZ,x0
n∏i=1
⟨V−ωi(0)
⟩D2
ZZ,x0
= 2πN e−1gs 2n+1δ
(ω −
n∑i=1
ωi
)sinh(πω)
n∏i=1
sinh(πωi).
(2.11)
Note that there is an a priori unknown normalization factor N in the integration measure
over the collective coordinate x0. In section 3, this normalization factor will be fixed by
comparison with the dual matrix model description2.
2.3 Next-to-leading order correction to the 1→ 1 amplitude
We now analyze the next-to-leading order one-instanton correction to the closed string am-
plitude, of order e−1/gsgs, focusing on the 1 → 1 case. As discussed in (2.4), it receives
contributions from the disc diagram with 2 closed string insertions, and from the discon-
nected diagram involving a disc and an annulus, each with 1 closed string insertion (Figure
2). Furthermore, there is a nontrivial cancelation of logarithmic divergences from individual
diagrams (which must be carefully regularized).
2.3.1 Disc 2-point diagram
In the first diagram of Figure 2, we take the incoming closed string vertex operator ccV+ω1
to
be fixed at z = i on the upper half plane, whereas the outgoing closed string vertex operatorc+c
2V−ω2
is fixed by the remaining conformal Killing vector to lie on the segment yi, y ∈ (0, 1).
The relevant free boson correlator on the upper half plane with Dirichlet boundary con-
dition X0|z=z = x0 is given by
CXD22
ω212
+ω222 y
ω222 (1− y)ω1ω2(1 + y)−ω1ω2ei(ω1−ω2)x0 , (2.12)
while the ghost correlator is
2CbcD2(1− y)(1 + y). (2.13)
Together with the Liouville correlator, the disc diagram gives (before integrating over the
2The normalization N will be given by a real constant factor associated with the Lorentzian scattering
amplitudes that we are computing. This is different from the Euclidean partition function of the ZZ-
instanton, which receives an imaginary contribution from the open string tachyon.
8
ZZ instanton collective coordinate)⟨V+ω1V−ω2
⟩D2
ZZ,x0
= 2g2sCD2ei(ω1−ω2)x02
ω212
+ω222
∫ 1
0
dy yω222 (1− y)1+ω1ω2(1 + y)1−ω1ω2
⟨Vω1
2(i)Vω2
2(yi)
⟩D2
Liouville,ZZ.
(2.14)
The Liouville correlator on the upper half plane with ZZ boundary condition can be evaluated
by integrating the appropriate structure constants multiplied by Virasoro conformal blocks
in either the “bulk channel”, where one first performs the OPE of the two bulk operators, or
the “boundary channel” where one first performs the bulk-to-boundary OPE. The relevant
conformal blocks are simply specializations of sphere 4-point Virasoro conformal blocks. The
bulk channel expression of the Liouville disc 2-point function is⟨Vω1
2(i)Vω2
2(yi)
⟩D2
Liouville,ZZ= (1 + y)−4−ω2
12ω212−ω
222 y
ω212−ω
222
∫ ∞0
dP
πC(ω1
2,ω2
2, P)
ΨZZ(P )
× F
(1 +
ω21
4, 1 +
ω22
4, 1 +
ω21
4, 1 +
ω22
4; 1 + P 2
∣∣∣∣∣(
1− y1 + y
)2),
(2.15)
where C(P1, P2, P3) is the DOZZ structure constant [26, 27] (following the convention of
[9, 11]), and F (h1, h2, h3, h4;h|z) is the c = 25 sphere 4-point Virasoro conformal block in
the 12 → 34 channel with external primaries of weight hi, i = 1, 2, 3, 4, internal primary of
weight h, and cross ratio z.
The boundary channel only involves the identity operator as the internal boundary pri-
mary. The result is expressed as⟨Vω1
2(i)Vω2
2(yi)
⟩D2
Liouville,ZZ=
A(1 + y)−4−ω212
ω212−ω
222 y
ω212−ω
222 ΨZZ
(ω1
2
)ΨZZ
(ω2
2
)F
(1 +
ω21
4, 1 +
ω21
4, 1 +
ω22
4, 1 +
ω22
4; 0
∣∣∣∣ 4y
(1 + y)2
),
(2.16)
where the numerical prefactor
A = 234π−
32 (2.17)
is fixed by crossing invariance between the bulk and boundary channels.3
3To derive (2.17), we can start from the crossing equation for the Liouville two-point function on the disk,
multiply both sides by the leg-pole factor S(ω1
2 )12 , where S(P ) ≡ − (Γ(2iP )/Γ(−2iP ))
2is the scattering
phase off the Liouville wall, and then analytically continue ω1 → 2i. In the latter limit, S(ω1
2 )12C(ω1
2 ,ω2
2 , P)
becomes proportional to δ(ω2
2 − P). The integral over P in (2.15) is then easily evaluated to give the value
of A in (2.16).
9
Let us analyze the limiting behavior of the y-integral in (2.14) near the boundary of the
moduli space. The integral near y = 1, using the bulk channel conformal block decomposition
of the Liouville correlator (2.15), behaves as∫ 1
dy
∫ ∞0
dP
π(1− y)−1+2P 2− (ω1−ω2)
2
2 C(ω1
2,ω1
2, P)
ΨZZ(P )
∼∫ ∞
0
dP
4P 2 − (ω1 − ω2)2C(ω1
2,ω1
2, P)
ΨZZ(P ).
(2.18)
Eventually we will integrate over the ZZ instanton collective coordinate x0 (see section
2.3.3), which enforces ω1 = ω2, in which case the y-integral converges near y = 1. Indeed,
C(ω1
2, ω1
2, P )ΨZZ(P ) ∼ P 2 for P ∼ 0, and so (2.18) converges near P = 0; likewise one can
verify that integration over large P in (2.18) converges as well.
Near y = 0, on the other hand, using the boundary channel conformal block decom-
position of the Liouville correlator (2.16), one finds that (2.14) has a divergent part given
by
g2sCD2
A
8ei(ω1−ω2)x0ΨZZ
(ω1
2
)ΨZZ
(ω2
2
)∫0
dy y−2(1− 2ω1ω2y). (2.19)
The leading power divergence is due to the open string “tachyon” on the ZZ-instanton,
and is simply subtracted in the usual regularization scheme. The subleading logarithmic
divergence, coming from the open string collective mode (represented by the boundary vertex
operator ∂X0), cannot be removed by any local counter term. Rather, it will cancel against
an analogous logarithmic divergence in the “annulus × disc” diagrams in Figure 4, to be
computed in section 2.3.2.
For now, we simply regularize the divergence by the subtraction4
⟨V+ω1V−ω2
⟩D2, reg
ZZ,x0= 2g2
sCD2ei(ω1−ω2)x02
ω212
+ω222
×∫ 1
0
dy
[yω222 (1− y)1+ω1ω2(1 + y)1−ω1ω2
⟨Vω1
2(i)Vω2
2(yi)
⟩D2
Liouville,ZZ
− A16
2−ω21+ω
22
2 ΨZZ(ω1
2
)ΨZZ
(ω2
2
)y−2(1− 2ω1ω2y)
].
(2.20)
To evaluate (2.20), we use the boundary channel representation of the Liouville correlator
(2.16) for 0 < y < 12, and the bulk channel representation (2.15) for 1
2< y < 1, and perform
the integration over the Liouville momentum P and the modulus y numerically. The details
are discussed in Appendix B.
4Note that the subtraction of power divergence here is slightly non-standard: the subtraction involves
the integral of y−2 over the range 0 < y < 1 rather than 0 < y < ∞. The difference between the two
subtraction schemes can be absorbed into the finite ambiguity discussed in section 2.3.3, which will be fixed
by comparison with the matrix model.
10
2.3.2 Annulus 1-point diagram
The second and third diagrams of Figure 2 are disconnected diagrams of a disc and an
annulus with 1 closed string insertion on each. The disc 1-point diagram is calculated as in
(2.8). We now turn to the annulus 1-point diagram.
We will parameterize the annulus as the strip 0 < Re(z) < π, with the identification
z ∼ z+ 2πit, t > 0. We take the closed string vertex operator to be V+ω , and both boundary
components of the annulus to end on the same ZZ instanton at x0. The diagram is evaluated
as ⟨V+ω
⟩A2
ZZ,x0= gs
∫ ∞0
dt
∫ 14
0
2πdx
⟨1
4π(b, µt) c2(2πx) : eiωX
0
: VP=ω2(2πx)
⟩A2(t)
ZZ,x0. (2.21)
The vertex operator is fixed to be on the real axis, at z = 2πx, accompanied by the ghost
insertion c2 ≡ c−c2i
. We further used the symmetry z → π − z to restrict the x-integration
range to (0, 14). The b ghost integrated against the Beltrami differential µt is given by
1
4π(b, ∂tg) = −π(b+ b). (2.22)
The ghost correlator is evaluated similarly to that of the holomorphic ghost system on the
torus via the doubling trick,⟨1
4π(b, µt)c2(2πx)
⟩A2(t)
= 2πCb,cA2η(it)2, (2.23)
where the overall phase is fixed by demanding that (2.23) is real and positive. The X0 part
of the correlator evaluates to⟨: e±iωX
0(2πx) :⟩A2(t)
= CX0
A2
e±iωx0
η(it)
[2π
∂νθ1(0|it)θ1 (2x|it)
]ω22
. (2.24)
Finally, the Liouville part of the annulus correlator can be expressed in terms of structure
constants and Virasoro conformal blocks in either the necklace channel or the OPE channel
(Figure 5). By the doubling trick, the relevant conformal blocks are restrictions of the torus
2-point conformal blocks considered in [9, 28].
In the necklace channel, we have⟨VP=ω
2(2πx)
⟩A2(t)
Liouviile,ZZ
= t−2−ω2
2
∫ ∞0
dP1
π
∫ ∞0
dP2
πC(ω
2, P1, P2
)ΨZZ(P1)ΨZZ(P2)q
h1− c24
1 qh2− c
242
× FNecklace
(q1, h1 = 1 + P 2
1 , d1 = 1 +ω2
4, q2, h2 = 1 + P 2
2 , d2 = 1 +ω2
4
),
(2.25)
11
P1 P2
(a)
11
(b)
Figure 5: Expansion of the Liouville bulk 1-point correlator on the annulus with ZZ boundary
conditions in (a) the necklace channel, and (b) the OPE channel.
where FNecklace(q1, h1, d1, q2, h2, d2) is the torus two-point c = 25 Virasoro conformal block in
the necklace channel, defined as in [28] on a torus of modulus τ = i/t that is a 2-fold cover
of the annulus. Here q1 = e−2πt
+ 4πxt and q2 = e−
4πxt are a pair of plumbing parameters, h1, h2
are the internal primary weights, d1, d2 are the external primary weights.
The OPE channel expression of the Liouville annulus 1-point function is given by a single
Virasoro conformal block,⟨VP=ω
2(2πx)
⟩A2(t)
Liouviile,ZZ
= AΨZZ(ω
2
)q−
c24 (2 sin(2πx))−2−ω
2
2 FOPE
(q, h1 = 0, v, h2 = 0, d1 = 1 +
ω2
4, d2 = 1 +
ω2
4
),
(2.26)
where FOPE(q, h1, v, h2, d1, d2) is the torus two-point OPE channel block defined in [28]. Here
the moduli are parameterized by q = e−2πt and v = 1−q2q2
, with q2 = e4πix. The numerical
constant A is the same as (2.17), as dictated by the crossing relation in the Liouville CFT
with ZZ boundary condition.
Putting these together, the annulus 1-point diagram is given by
⟨V+ω
⟩A2
ZZ,x0= 4π2gsCA2
∫ ∞0
dt η(it)
∫ 14
0
dx
[2π
∂νθ1(0|it)θ1 (2x|it)
]ω22
eiωx0〈VP=ω
2(2πx)〉A
2(t)Liouviile,ZZ.
(2.27)
where 〈VP=ω2(2πx)〉A
2(t)Liouviile,ZZ can be evaluated using either (2.25) or (2.26).
Let us study the limiting behavior of the t, x integral near the boundary of the moduli
12
space. For t away from 0 and ∞, the integral near x = 0 takes the form
π2gsCA2Aeiωx0
ΨZZ(ω
2
)∫dt
∫0
dxe2πt − 1
sin2(2πx)
[1 +O(x2)
]. (2.28)
Here we see a power divergence associated to the open string tachyon mode on the ZZ
instanton. We will regularize it by simply subtracting the term proportional to 1sin2(2πx)
in
the x-integrand. Possible finite ambiguities in this regularization scheme will be discussed
in section 2.3.3. Note that there is no logarithmic divergence near x = 0.
The integral near t =∞ takes the form
π2gsCA2Aeiωx0
ΨZZ(ω
2
)∫ ∞dt
∫ 14
0
dx
[e2πt − 1
sin2(2πx)+ 2ω2
]. (2.29)
After subtracting off the power divergence in the x-integral near x = 0, as discussed above,
we end up with a subleading divergence of the form
π2
2gsCA2Aω
2eiωx0
ΨZZ(ω
2
)∫ ∞dt. (2.30)
Comparing this limit to the disc 1-point amplitude also allows us to fix the normalization
constant
CA2 = 1. (2.31)
After multiplying by the disc 1-point diagram⟨V−ω′(z, z)
⟩D2
ZZ,x0(as given by (2.8)), and in-
tegrating over the ZZ-instanton collective coordinate x0, this divergence will cancel against
the logarithmic divergence in the disc 2-point diagram (2.19), provided that we make the
identification
y ∼ e−2πt (2.32)
near y = 0 and t =∞ respectively. In view of Figure 4, the degeneration limits of both the
disc 2-point diagram and the disconnected disc×annulus diagram can be viewed as a pair
of discs attached to the same thin strip in different ways. (2.32) is then indeed the correct
identification of moduli in the pinching limit, up to a finite constant ambiguity (multiplicative
in y, or additive in t) that cannot be determined by the argument given so far.
Finally, let us consider integration near t = 0. As discussed in appendix A, for real ω
there are potential divergences due to the propagation of on-shell closed string states that
need to be regularized. A simple way to treat this, as in [9], is to analytically continue
from ωi = ω purely imaginary, where no such regularization is needed. Such an analytic
continuation is straightforward in the domain 0 < Im(ω) < 1 on the complex ω-plane, where
no poles cross the integration contour over internal Liouville momenta. One may then take
the real ω limit to obtain the physical amplitude.
13
For the sake of comparison with the matrix model dual, we will hereafter work with
purely imaginary ω in the range 0 < Im(ω) < 1, where the regularized annulus amplitude
takes the form
⟨V+ω
⟩A2
ZZ,x0= 4π2gsCA2eiωx
0
∫ ∞0
dt
∫ 14
0
dx
η(it)
[2π
∂νθ1(0|τ)θ1 (2x|τ)
]ω22 ⟨
VP=ω2(2πx)
⟩A2(t)
Liouville,ZZ
− A
4ΨZZ
(ω2
)[ e2πt − 1
sin2(2πx)+ 2ω2
].
(2.33)
The details of the numerical evaluation of (2.33) are described in Appendix B.
2.3.3 Integration over collective coordinate and cancelation of divergences
We can now put together the worldsheet diagrams in Figure 2 and integrate over the ZZ-
instanton collective coordinate x0 as in (2.4) to obtain the contribution to S1→1 at order
e−1/gsgs. As already discussed, the divergences in each diagram due to the exchange of open
string collective mode cancel. It suffices to compute the finite part of the diagrams according
to the regularized expressions (2.20) and (2.33). There are, however, two possible further
corrections at this order.
The first correction is due to the renormalization of SZZ in (2.4). The expression e−SZZ
with SZZ = 1gs
can be thought of as summing over empty disc diagrams. At subleading orders
in gs, there will be contributions from annulus diagrams of order g0s , followed by two-holed
discs of order gs, and so forth. Formally, the cylinder diagram renormalizes the measure
factor N in the integration over the collective coordinate, e.g. in (2.11). The two-holed disc
effectively shifts SZZ by gsS(1)ZZ , so that the order e−1/gsgs contribution to S1→1 is corrected
by a term proportional to the “disc 1-point squared” amplitude,
−e−1/gsgsS(1)ZZ 2πN δ(ω1 − ω2)
[gsCD2
2πΨZZ
(ω1
2
)]2
. (2.34)
However, these diagrams without closed string insertions are divergent due to open string
tachyons, and the regularization of the divergence is subject to finite ambiguities similar to
the one encountered in the annulus 1-point diagram in section 2.3.2. Rather than trying
to regularize all of these divergences simultaneously in a consistent manner, we will simply
assume that S(1)ZZ in (2.34) is a constant, and will fix it along with N by comparing with the
proposed matrix model dual.
The second correction to (2.4) has to do with the ambiguity in cutting off the logarithmic
divergences due to the open string collective mode in the pinching limit of Figure 4. Namely,
14
in identifying the cutoff parameters on the moduli y versus t according to (2.32), there is a
finite constant ambiguity. This leads to an ambiguity in the order e−1/gsgs contribution to
S1→1 of the form
c′2πN δ(ω1 − ω2)g2sCD2
A
4ω2
1
[ΨZZ
(ω1
2
)]2
. (2.35)
We do not know of a worldsheet prescription that fixes c′ a priori. It will be determined by
comparison with the matrix model dual as well.
Altogether, the next-to-leading order one-instanton correction to the 1→ 1 closed strings
amplitude is given by
Sinst,(1)1→1 (ω;ω′)
= e−1/gsgsδ(ω − ω′)8πN
π
12 2−
14
+ω2
∫ 1
0
dy
[yω2
2 (1− y)1+ω2
(1 + y)1−ω2 ⟨Vω
2(i)Vω
2(yi)
⟩D2
Liouville,ZZ
− A
162−ω
2(
ΨZZ(ω
2
))2
y−2(1− 2ω2y)
]+ 2
34π
32 ΨZZ
(ω2
)∫ ∞0
dt
∫ 14
0
dx
[η(it)
(2π
∂νθ1(0|τ)θ1 (2x|τ)
)ω2
2 ⟨VP=ω
2(2πx)
⟩A2(t)
Liouville,ZZ
− A
4ΨZZ
(ω2
)( e2πt − 1
sin2(2πx)+ 2ω2
)]
− S(1)ZZ sinh2(πω) + c′ ω2 sinh2(πω)
.
(2.36)
3 Comparison with the matrix model dual
The matrix model dual to the closed string sector of c = 1 string theory is defined as a suit-
able N →∞ scaling limit of the U(N) gauged matrix quantum mechanics with Hamiltonian
H = 12Tr(P 2−X2), where X is a Hermitian N ×N matrix and P the canonically conjugate
momentum matrix. Upon writing X = Ω−1ΛΩ, where Ω ∈ U(N) and Λ = diag(λ1, · · · , λN)
a diagonal matrix, the wave function which is restricted to be U(N)-invariant can be ex-
pressed as a function of the eigenvalues Ψ(Λ) that is completely symmetric in the λi’s.
The Hamiltonian can be put to the form H = ∆−1H ′∆, where ∆ =∏
i<j(λi − λj) and
H ′ = 12
∑i(−∂2
λi−λ2
i ) is the Hamiltonian of N free non-relativistic particles in the potential
V (λ) = −12λ2. As H ′ effectively acts on the wave function Ψ′ = ∆Ψ, which is completely
anti-symmetric in the λi’s, the eigenvalues are coordinates of free fermions.
15
Perturbatively, the closed string vacuum of c = 1 string theory is dual to the state of
the matrix model in which the fermions reside on the right side of the potential (x > 0)
and occupy all states up to the fermi energy −µ, µ > 0. The string coupling is related
by 2πgs = µ−1. Closed string states are dual to collective excitations of the fermi surface.
Following the formalism of [6], the collective modes can be represented as those of particle-
hole pairs, and the S-matrix can be computed as a sum/integral over reflection amplitudes
of individual particles and holes.
Non-perturbatively, the above prescription of the matrix model ground state is ill-defined.
Rather than speaking of the fermions filling regions of the phase space, which only makes
sense semi-classically, we will consider exact energy eigenstates of the fermion, which are
scattering states of the Hamiltonian H = 12p2− 1
2x2. At a given energy E, there is a two-fold
degeneracy: |E〉R which describes a scattering state with no incoming flux from the left side
of the potential, and |E〉L which has no incoming flux from the right side. We propose that
the non-pertubative closed string vacuum of c = 1 string theory is dual to the state of the
matrix model in which all scattering states of the form
|E〉R, E ≤ −µ, (3.1)
and none other, are occupied by free fermions. Clearly, in the semi-classical limit (large µ)
this reduces to the picture of fermions filling the right fermi sea.
The exact reflection amplitude of a single fermion in frequency space is given by [6]5
R(E) = iµiE[
1
1 + e2πE·
Γ(12− iE)
Γ(12
+ iE)
] 12
. (3.2)
To compute the S-matrix of the collective modes, we also need to know the reflection am-
plitude of a hole. The “hole scattering state” amounts to the un-occupation of the fermion
scattering state |E〉R. It describes the hole moving in the opposite direction of the would-be
fermion in the state |E〉R, with the roles of the incoming and outgoing waves exchanged.
From this we deduce that the reflection amplitude of a hole is the inverse of that of a fermion,
namely (R(E))−1. While the fermion reflection amplitude R(E) has magnitude less than 1
due to transmission/tunneling, the reflection amplitude of a hole has magnitude greater than
1. Note that this is different from the type 0B matrix model [23,24], or the theory of “type
II” in [6], where both sides of the fermi sea are filled and the reflection amplitude of a hole
is (R(E))∗.
5Note that R(E) has poles on the lower half complex E-plane, as expected of S-matrix elements of
non-relativistic scattering off a potential barrier.
16
The exact 1→ n scattering amplitude is then given by [6]6
A1→n(ω1, · · · , ωn) = −∑
S1tS2=S
(−1)|S2|∫ ω(S2)
0
dxR(−µ+ ω − x)(R(−µ− x))−1, (3.3)
where S1, S2 are disjoint subsets of S = ω1, ..., ωn such that S1 t S2 = S. |S2| denotes
the number of elements of S2, and ω(S2) the sum of all elements of S2. We can write the
integrand of (3.3) as
R(−µ+ ω − x)(R(−µ− x))−1 =
[1 + e−2πµe−2πx
1 + e−2πµe2π(ω−x)
] 12
K(µ, ω, x), (3.4)
where
K(µ, ω, x) ≡ µiω[
Γ(12− i(−µ+ ω − x))
Γ(12− i(−µ− x))
Γ(12
+ i(−µ− x))
Γ(12
+ i(−µ+ ω − x))
] 12
. (3.5)
The asymptotic series expansion of K(µ, ω, x) in µ−1, integrated in x according to (3.3), gives
the perturbative expansion of the matrix model amplitude that is expected to agree with the
usual perturbative closed string amplitudes of c = 1 string theory. Note that K(µ, ω, x) is
not analytic in µ at µ =∞. Nonetheless, one can verify that the “perturbative part” of the
amplitude, produced by the x-integral of K(µ, ω, x), obeys the criteria of [29]7 and is thus
given by the Borel summation of the perturbative series by a theorem of Nevanlinna [30].
The non-perturbative corrections comes from the prefactor on the RHS of (3.4). We can
expand (3.4) as
R(−µ+ ω − x)(R(−µ− x))−1
= K(µ, ω, x)− e−2πµeπ(ω−2x) sinh(πω)
[1−
iω(x− ω2)
µ+O(µ−2)
]+O(e−4πµ).
(3.6)
The term of order e−2πµ = e−1/gs gives rise to the leading non-perturbative contribution to
the 1→ n amplitude
Ainst,(0)1→n (ω1, · · · , ωn) = −2n+1 e
−2πµ
4πsinh(πω)
n∏i=1
sinh(πωi). (3.7)
This is in precise agreement with the worldsheet amplitude of disconnected discs with bound-
aries ending on a single ZZ-instanton (2.11), up to the normalization constant N in the
6In comparison to the analogous formulae in [6], R∗ is replaced by R−1 here.7This follows from the fact that K(µ, ω, x) is analytic on the open domain Reµ > Λ for some sufficiently
large positive Λ (at fixed ω and x), approaching 1 as µ→∞, and that the error of the asymptotic series of
logK in µ−1 truncated to order µ−N in this domain is uniformly bounded by CN !(σ/µ)N for some positive
constants C and σ, as can seen from the integral representation of log-Gamma function.
17
integration measure of the ZZ-instanton collective coordinate. Identifying the two results
determines
N = − 1
8π2. (3.8)
The term of order e−2πµµ on the RHS of (3.6), upon integration in x, gives the following
next-to-leading order non-perturbative contribution to the 1→ 1 amplitude
Ainst,(1)1→1 (ω) = −ie
−2πµ
2π2µω
(πω
tanh(πω)− 1
)sinh2(πω). (3.9)
0.2 0.4 0.6 0.8 1.0Im(ω)
0.1
0.2
0.3
0.4
A1->1inst,(1)
Figure 6: In red dots is shown the non-perturbative correction of order e−1/gsgs to the 1→ 1
closed string amplitude, computed from the worldsheet by numerically evaluating (2.36)
with the identification (3.10). In blue the matrix model result (3.9) for purely imaginary ω
is plotted.
(3.9) can be compared to the worldsheet computation of disc 2-point amplitude and disc
× annulus amplitude, which is given by the analytic continuation of the expression (2.36)
which is a priori valid for purely imaginary ω with 0 < Im(ω) < 1. Note that for purely
imaginary ω, both (2.36) and (3.9) are manifestly real. (2.36) is evaluated numerically in
this domain, up to the so-far-undetermined constants S(1)ZZ and c′. We find an excellent fit to
(3.9) with8
S(1)ZZ = 0.496± 0.001, c′ = −1.399± 0.002. (3.10)
The striking agreement between the worldsheet and matrix model results, provided the
identification (3.10), is shown in Figure 6.
8Here we have only indicated the error bar of the least squares fit, without taking into account the
numerical error in the worldsheet computation itself.
18
The agreement of the ZZ-instanton computation with the non-perturbative corrections
in the matrix model amplitude distinguishes our non-perturbative completion of the matrix
model dual from earlier proposals in the literature. For instance, as already mentioned, type
0B matrix model which amounts to filling both sides of the fermi sea up to E ≤ −µ would
lead to the reflection amplitude (R(E))∗ for the hole, rather than (R(E))−1 [6, 23, 24], and
would lead to different ω-dependence in the order e−2πµ amplitudes. Similarly, the theory
of “type I” considered in [6], in which the reflection amplitude R(E) is a phase, would also
lead to different answers from the ZZ-instanton computation.9
4 Discussion
In this paper we studied non-perturbative effects mediated by ZZ instantons in the world-
sheet formulation of c = 1 string theory, in which the Fischler-Susskind-Polchinski (FSP)
mechanism plays an essential role in canceling open string divergences. We found that certain
one-instanton amplitudes of closed strings at the leading and next-to-leading orders in gs are
in agreement with the matrix model dual, provided that the closed string vacuum is identified
with the state in the matrix quantum mechanics in which all scattering energy levels with
no incoming flux from the other side of the potential are filled by free fermions/eigenvalues.
Our proposal for the non-perturbative completion of c = 1 string theory and its dual matrix
model is explicitly different from that of type 0B string theory in two dimensions.
In our worldsheet computation of the 1 → 1 closed string amplitude at order e−1/gsgs,
there are two unfixed constants. One of them, S(1)ZZ , is formally computed by the worldsheet
diagram of a 2-holed disc with no closed string insertions, which amounts to integrating
a genus two Virasoro vacuum conformal block over a suitable moduli space. The latter is
divergent near the boundary of the moduli space, due to the open string tachyon rather
than the collective mode. It may be possible to regularize this divergence carefully so as
to be compatible with the regularization of open string tachyon divergence of the annulus
amplitude in section 2.3.2, and give a worldsheet derivation of the prediction (3.10) from
the matrix model dual. The other constant c′ arises as a finite ambiguity in the cancelation
of logarithmic divergences between diagrams of different topologies via the FSP mechanism.
While c′ is eventually fixed by matching with the matrix model result, we do not know an a
priori prescription for determining c′ in the worldsheet formalism.10
Note that while the perturbative expansion of the closed string amplitude in c = 1
string theory is an asymptotic series, it is Borel summable, and the result of the Borel
9In fact, the theory of “type I” has been argued in [31] to be non-perturbatively inconsistent with c = 1
string theory, due to causality constraints on the scattering of “tall” pulses.10A similar ambiguity in the D-instanton effects in type IIB superstring theory was noted in [20].
19
summation produces the “perturbative part” of the amplitude captured by the function K
in (3.4). The ZZ-instanton effect gives rise to non-perturbative corrections on top of this.
It will be interesting to see whether the multi-ZZ-instanton expansion reproduces the full
non-perturbative amplitude of the matrix model, captured by the prefactor on the RHS
of (3.4). It would also be desirable to recast the ZZ-instanton amplitudes in the language
of open+closed string field theory [32], in such a way that the computation is manifestly
finite, without the need for integration near the boundary of the moduli space. In doing
so, one would presumably encounter finite ambiguities analogous to c′ in FSP cancelation
mechanism at higher orders. A key question is how to fix these ambiguities based on the
non-perturbative consistency of the string theory.11
The non-perturbative (exclusive) closed string amplitude, as we have seen, is not unitary.
The unitary completion requires taking into account other asymptotic states, particularly
those of fermions on the other side of the potential. The latter correspond to ZZ-branes
in c = 1 string theory with open string tachyon rolling to the “wrong side” of the tachyon
potential. It is conceivable that scattering amplitudes of closed string and ZZ-branes with
rolling tachyon can be computed by worldsheet diagrams with ZZ-brane (rather than ZZ-
instanton) boundary conditions, but a systematic formulation thereof is yet to be developed.
Acknowledgements
We would like to thank Igor Klebanov, Juan Maldacena, Ashoke Sen, Marco Serone, Steve
Shenker, Andy Strominger, and Yifan Wang for discussions. BB and VR thank the organizers
of Bootstrap 2019 at Perimeter Institute, VR thanks the organizers of Strings 2019, Brussels,
XY thanks the Galileo Galilei Institute for Theoretical Physics, Florence, the Center for
Quantum Mathematics and Physics at University of California, Davis, International Center
for Theoretical Physics, Trieste, and Nordic Institute for Theoretical Physics, Stockholm, for
their hospitality during the course of this work. This work is supported in part by a Simons
Investigator Award from the Simons Foundation, by the Simons Collaboration Grant on
the Non-Perturbative Bootstrap, and by DOE grant de-sc00007870. BB is supported by
the Bolsa de Doutoramento FCT fellowship. VR is supported by the National Science
Foundation Graduate Research Fellowship under Grant No. DGE1144152.
11A possible clue lies in the matrix model result (3.3): the non-perturbative correction to the fermion
reflection phase R(E) in (3.2) is responsible for turning branch cuts on the complex E-plane into poles, as
required by general properties of non-relativistic scattering, and cannot be modified arbitrarily (assuming
that the dual description is a system of free fermions).
20
A The t→ 0 limit of the annulus 1-point diagram
In the t→ 0 limit, the annulus 1-point diagram has potential divergences due to propagation
of on-shell closed string states that must be regularized. Changing the variable t = 1/s and
using the modular property of η and θ1, we can rewrite (2.27) as⟨V+ω
⟩A2
ZZ,x0
= −4π2gsCA2eiωx0
∫ ∞ds s−
32η(is)
∫ 14
0
dx
[2πe−4πsx2
is∂νθ1(0|is)θ1(2isx|is)
]ω22 ⟨
VP=ω2(2πx)
⟩A2(1/s)
Liouviile,ZZ.
(A.1)
In the s → ∞ limit, we have 2π∂νθ1(0|is)θ1(2isx|is) → 2 sinh(2πxs), and the Liouville annulus
1-point function reduces to an integral of disc 2-point functions (for 0 < x < 14)⟨
VP=ω2(2πx)
⟩A2(1/s)
Liouviile,ZZ
→ s2+ω2
2
∫ ∞0
dP1
πΨZZ(P1)e−2πs(P 2
1−124) (2 tanh(πsx))2+2P 2
1
(sinh(2πsx))2+ω2
2
⟨VP=ω
2(i)VP1(yi)
⟩D2
Liouviile,ZZ,
(A.2)
where y ≡ tanh(πxs). Altogether, the large s contribution to (A.1) can be written as
⟨V+ω
⟩A2
ZZ,x0∼ −4π2gsCA2eiωx
0
2ω2
2
∫ ∞ds s
12
∫ 14
0
dxe−2πsx2ω2
sinh2(2πxs)
×∫ ∞
0
dP1
πΨZZ(P1)e−2πsP 2
1 (2 tanh(πxs))2+2P 21⟨VP=ω
2(i)VP1(iy)
⟩D2
Liouviile,ZZ.
(A.3)
The divergence comes from singular limits of the disc 2-point function on the RHS. The
divergence in the small y or small xs limit was already analyzed in (2.28). There is also a
potential divergence coming from the y → 1 or large xs limit, where
⟨V+ω
⟩A2
ZZ,x0∼ −4π2gsCA2eiωx
0
∫ ∞dss
12
∫ 14
0
dx
∫ ∞0
dP1dP2
π2ΨZZ(P1)ΨZZ(P2)C
(ω2, P1, P2
)× e−2πs[(1−2x)P 2
1 +2xP 22−x( 1
2−x)ω2].
(A.4)
In particular, if Re(ω2) > 0, the s-integral diverges for sufficiently small P1 and P2, and
must be regularized in a way that preserves the expected analyticity in ω. For numerical
evaluation and comparison to the matrix model results, it is convenient to work with purely
imaginary ω, in which case the s-integral is manifestly convergent.
In the analytic continuation in ω, one must ensure that the poles of the DOZZ structure
constant, located at ±ω2
+ (P1±P2) = in for nonzero integer n, do not cross the integration
21
contour in P1 and P2, or equivalently, take into account of residue contributions when a pole
crosses the integration contour. The simplest regime is 0 < Im(ω) < 1, where such residue
contributions are absent, and the regularized amplitude is given by (2.33).
B Some numerical details
B.1 Disc 2-point diagram
The disc 2-point Virasoro conformal blocks appearing in (2.15) and (2.16) are evaluated as
an expansion in the elliptic nome q = exp[−πK(1−η)/K(η)], where η is the cross-ratio of the
correlator and K(η) = 2F1(1/2, 1/2, 1; η), using Zamolodchikov’s recurrence relations [33].
Truncating the q-expansion to relatively low orders (∼ 10 − 20) is sufficient to obtain very
accurate results.
0.5 1.0 1.5 2.0 2.5P
-1.5
-1.0
-0.5
<Vω/2Vω/2>D2 ,bulk-ch(P)
Figure 7: A sample plot of the P -integrand in (2.20) from expanding the Liouville correlator
in the bulk channel, after performing the y-integral for 1/2 < y < 1. The energy of the
incoming tachyon is taken to be ω = 0.4i.
The numerical integration over the moduli space of the 2-punctured disc (2.20) is split into
two regions: 0 < y < 12, where the boundary channel conformal block representation is used,
and 12< y < 1 where the bulk channel representation is used. In the boundary channel, only
the vacuum channel block contributes, and the y-integral is straightforwardly evaluated after
the subtraction in (2.20). In the bulk channel, the conformal block decomposition involves an
integration over the Liouville momentum P . It is convenient to first perform the y-integral
and then the P -integral. In the vicinity of y = 1, we can truncate the Virasoro conformal
22
block to the lowest order in its q-expansion and perform the integration analytically as a
function of P . The remaining of integration over 1/2 < y < 1 is evaluated numerically at a
sufficiently large set of sample values of P . We then numerically interpolate in P and perform
the P -integral. A sample plot of the P -integrand is shown in Figure 7. The regularized disc
2-point diagram (2.20), after integrating the collective coordinate, evaluated over a set of
imaginary ω with 0 < Im(ω) < 1, is shown in Figure 8.
0.2 0.4 0.6 0.8Im(ω)
-0.1
0.1
0.2
0.3
0.4
disc 2-pt(disc 1-pt) x (annulus 1-pt)
Figure 8: Numerical results for the disc 2-point diagram (in blue) and the (disc 1-
point)×(annulus 1-point) diagram (in orange) contributions to (2.36) (with the prefactor
e−1/gsgsδ(ω − ω′) stripped off), at imaginary values of ω in the range 0 < Im(ω) < 1.
B.2 Annulus 1-point diagram
For the annulus 1-point diagram, we divide the moduli space into a number of domains. In
each domain, a specific Virasoro conformal block decomposition and approximation scheme
is used, as indicated in Figure 9. Sample plots of the integrand in various domains are shown
in Figure 10.
For t1 = 0.3 ≤ t ≤ t2 = 1, we express the Liouville annulus 1-point function in (a) the
OPE channel (2.26) for ε = 0.005 ≤ x ≤ x∗ = 0.1, and (b) the necklace channel (2.25) for
x∗ < x ≤ 1/4. In each channel, the annulus 1-point conformal block, which is a special case
of the torus 2-point block, is evaluated using the recurrence relations of [28] and truncated
in their expansion in the appropriate plumbing parameters (similarly to the evaluation of
the genus one amplitude in [9]). In the domain (c) t2 ≤ t ≤ t3 = 2, it is convenient to
use the OPE channel representation in which only the vacuum conformal block contributes.
Finally, for sufficiently large t the integrand becomes negligibly (exponentially) small and
we can truncate the integration at t = t3.
23
t
xt1
t2
ε x∗ 1/4
t3
(a)
OPE
channel
(b)
necklace
channel
(c) OPE channel
negligible
(d)
s = 1/t
x1/t1
s2
ε x∗ 1/4
s1
(e)
boundary
channel(f) bulk channel
(g) asymptotics (A.4)
Figure 9: Division of the moduli space of the 1-punctured annulus, for the purpose of
numerical integration. The left figure covers the t > t1 region whereas the right figure covers
the t < t1 region.
For smaller values of t ≤ t1 we can approximate the moduli integrand with the disc
2-point correlator as in (A.3). After passing to the variable s = 1/t, we divide the (x, s)
integration domain according to the RHS of Figure 9. In region (f) with x∗ < x ≤ 1/4 and
1/t1 < s < s1 = 10, we expand the Liouville disc correlator in the bulk channel, whereas in
region (e) with 1/t1 < s < 1/x and ε ≤ x ≤ x∗ we express the disc correlator in the boundary
channel, using Zamolodchikov’s recurrence relation and truncate in the q-series as before.
In the region (g), one can further approximate the integrand using (A.4) and perform the
s-integral over s > maxs1, 1/x analytically.
Finally, in the slab region (d) x < ε, the integrand becomes negligibly small whenever
t > t2 or s > s2. For intermediate values of t or s we simply perform a linear extrapolation
of the data points of adjacent regions, namely (a) and (e), down to x = 0.
Collecting contributions from all of these domains of the moduli space, the result for the
regularized annulus 1-point diagram (2.33), evaluated over a set of imaginary ω, is shown in
Figure 8.
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0 0.05 0.10 0.15 0.20 0.25
0.4
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